Amplitude equations for SPDEs with quadratic nonlinearities forced by additive and multiplicative noise

TL;DR

This paper derives amplitude equations for SPDEs with quadratic nonlinearities under small additive and multiplicative noise, providing rigorous error analysis and applying results to stochastic Burger's equation.

Contribution

It introduces a method to approximate complex SPDEs with simpler amplitude equations and offers a rigorous error estimate for this approximation.

Findings

Successful derivation of amplitude equations for the class of SPDEs considered.

Rigorous error bounds established for the approximation.

Application demonstrated on stochastic Burger's equation.

Abstract

This article deals with stochastic partial differential equations with quadratic nonlinearities perturbed by small additive and multiplicative noise. We present the approximate solution of the original equation via the amplitude equation and give the rigorous error analysis. For illustration, we apply our main theorems to stochastic Burger's equation.